YES(O(1), O(n^2)) 39.72/18.93 YES(O(1), O(n^2)) 39.72/18.95 39.72/18.95 39.72/18.95 39.72/18.95 39.72/18.95 39.72/18.95 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 39.72/18.95 39.72/18.95 39.72/18.95
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
39.72/18.95 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳2 CdtProblem
39.72/18.95 
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳4 CdtProblem
39.72/18.95 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳6 CdtProblem
39.72/18.95 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳8 CdtProblem
39.72/18.95 
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳10 CdtProblem
39.72/18.95 
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳12 CdtProblem
39.72/18.95 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
39.72/18.95 
↳14 CdtProblem
39.72/18.95 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
39.72/18.95 
↳16 CdtProblem
39.72/18.95 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳18 CdtProblem
39.72/18.95 
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳20 CdtProblem
39.72/18.95 
↳21 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳22 CdtProblem
39.72/18.95 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
39.72/18.95 
↳24 CdtProblem
39.72/18.95 
↳25 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳26 CdtProblem
39.72/18.95 
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
39.72/18.95 
↳28 CdtProblem
39.72/18.95 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
39.72/18.95 
↳30 CdtProblem
39.72/18.95 
↳31 SIsEmptyProof (BOTH BOUNDS(ID, ID))
39.72/18.95 
↳32 BOUNDS(O(1), O(1))
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39.72/18.95

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

le(0, y) → true 39.72/18.95
le(s(x), 0) → false 39.72/18.95
le(s(x), s(y)) → le(x, y) 39.72/18.95
minus(x, 0) → x 39.72/18.95
minus(0, x) → 0 39.72/18.95
minus(s(x), s(y)) → minus(x, y) 39.72/18.95
gcd(0, y) → y 39.72/18.95
gcd(s(x), 0) → s(x) 39.72/18.95
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) 39.72/18.95
if_gcd(true, x, y) → gcd(minus(x, y), y) 39.72/18.95
if_gcd(false, x, y) → gcd(minus(y, x), x)

Rewrite Strategy: INNERMOST
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39.72/18.95

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
39.72/18.95
39.72/18.95

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 39.72/18.95
le(s(z0), 0) → false 39.72/18.95
le(s(z0), s(z1)) → le(z0, z1) 39.72/18.95
minus(z0, 0) → z0 39.72/18.95
minus(0, z0) → 0 39.72/18.95
minus(s(z0), s(z1)) → minus(z0, z1) 39.72/18.95
gcd(0, z0) → z0 39.72/18.95
gcd(s(z0), 0) → s(z0) 39.72/18.95
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 39.72/18.95
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 39.72/18.95
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.95
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 39.72/18.95
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.95
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.95
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 39.72/18.95
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.95
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c9, c10

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39.72/18.95

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) by

GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)), LE(0, z0)) 39.72/18.95
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))), LE(s(z0), 0)) 39.72/18.95
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
39.72/18.95
39.72/18.95

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 39.72/18.95
le(s(z0), 0) → false 39.72/18.95
le(s(z0), s(z1)) → le(z0, z1) 39.72/18.95
minus(z0, 0) → z0 39.72/18.95
minus(0, z0) → 0 39.72/18.95
minus(s(z0), s(z1)) → minus(z0, z1) 39.72/18.95
gcd(0, z0) → z0 39.72/18.95
gcd(s(z0), 0) → s(z0) 39.72/18.95
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 39.72/18.95
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 39.72/18.95
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.95
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.95
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.95
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)), LE(0, z0)) 39.72/18.95
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))), LE(s(z0), 0)) 39.72/18.95
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.95
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.96
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.96
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.96
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)), LE(0, z0)) 39.72/18.96
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))), LE(s(z0), 0)) 39.72/18.96
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c9, c10, c8

39.72/18.96
39.72/18.96

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
39.72/18.96
39.72/18.96

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 39.72/18.96
le(s(z0), 0) → false 39.72/18.96
le(s(z0), s(z1)) → le(z0, z1) 39.72/18.96
minus(z0, 0) → z0 39.72/18.96
minus(0, z0) → 0 39.72/18.96
minus(s(z0), s(z1)) → minus(z0, z1) 39.72/18.96
gcd(0, z0) → z0 39.72/18.96
gcd(s(z0), 0) → s(z0) 39.72/18.96
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 39.72/18.96
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 39.72/18.96
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.96
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.96
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.96
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.96
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 39.72/18.96
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 39.72/18.96
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.96
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.96
IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) 39.72/18.96
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.96
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 39.72/18.96
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 39.72/18.96
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c9, c10, c8, c8

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39.72/18.96

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF_GCD(true, z0, z1) → c9(GCD(minus(z0, z1), z1), MINUS(z0, z1)) by

IF_GCD(true, z0, 0) → c9(GCD(z0, 0), MINUS(z0, 0)) 39.72/18.96
IF_GCD(true, 0, z0) → c9(GCD(0, z0), MINUS(0, z0)) 39.72/18.96
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
39.72/18.96
39.72/18.96

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 39.72/18.96
le(s(z0), 0) → false 39.72/18.96
le(s(z0), s(z1)) → le(z0, z1) 39.72/18.96
minus(z0, 0) → z0 39.72/18.96
minus(0, z0) → 0 39.72/18.97
minus(s(z0), s(z1)) → minus(z0, z1) 39.72/18.97
gcd(0, z0) → z0 39.72/18.97
gcd(s(z0), 0) → s(z0) 39.72/18.97
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 39.72/18.97
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 39.72/18.97
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.97
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.97
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.97
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 39.72/18.97
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 39.72/18.97
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 39.72/18.97
IF_GCD(true, z0, 0) → c9(GCD(z0, 0), MINUS(z0, 0)) 39.72/18.97
IF_GCD(true, 0, z0) → c9(GCD(0, z0), MINUS(0, z0)) 39.72/18.97
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 39.72/18.97
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 39.72/18.97
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 39.72/18.97
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 39.72/18.97
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 39.72/18.97
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, z0, 0) → c9(GCD(z0, 0), MINUS(z0, 0)) 40.21/19.01
IF_GCD(true, 0, z0) → c9(GCD(0, z0), MINUS(0, z0)) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c10, c8, c8, c9

40.21/19.01
40.21/19.01

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
40.21/19.01
40.21/19.01

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.01
le(s(z0), 0) → false 40.21/19.01
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.01
minus(z0, 0) → z0 40.21/19.01
minus(0, z0) → 0 40.21/19.01
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.01
gcd(0, z0) → z0 40.21/19.01
gcd(s(z0), 0) → s(z0) 40.21/19.01
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.01
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.01
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.01
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.01
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.01
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.01
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.01
IF_GCD(true, z0, 0) → c9 40.21/19.01
IF_GCD(true, 0, z0) → c9
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.01
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.01
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.01
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.01
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.01
IF_GCD(true, z0, 0) → c9 40.21/19.01
IF_GCD(true, 0, z0) → c9
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c10, c8, c8, c9, c9

40.21/19.01
40.21/19.01

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

IF_GCD(true, 0, z0) → c9 40.21/19.01
IF_GCD(true, z0, 0) → c9
40.21/19.01
40.21/19.01

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.01
le(s(z0), 0) → false 40.21/19.01
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.01
minus(z0, 0) → z0 40.21/19.01
minus(0, z0) → 0 40.21/19.01
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.01
gcd(0, z0) → z0 40.21/19.01
gcd(s(z0), 0) → s(z0) 40.21/19.01
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.01
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.01
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.01
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.01
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.01
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.01
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.01
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.01
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.01
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.01
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c10, c8, c8, c9

40.21/19.01
40.21/19.01

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:

minus(z0, 0) → z0 40.21/19.01
minus(0, z0) → 0 40.21/19.01
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.01
le(0, z0) → true 40.21/19.01
le(s(z0), 0) → false 40.21/19.01
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.01
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.01
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.01
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.01
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.01
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.01
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 40.21/19.01

POL(0) = 0    40.21/19.01
POL(GCD(x1, x2)) = [4]x1 + [4]x2    40.21/19.01
POL(IF_GCD(x1, x2, x3)) = [4]x2 + [4]x3    40.21/19.01
POL(LE(x1, x2)) = 0    40.21/19.01
POL(MINUS(x1, x2)) = 0    40.21/19.01
POL(c10(x1, x2)) = x1 + x2    40.21/19.01
POL(c2(x1)) = x1    40.21/19.01
POL(c5(x1)) = x1    40.21/19.01
POL(c8(x1)) = x1    40.21/19.01
POL(c8(x1, x2)) = x1 + x2    40.21/19.01
POL(c9(x1, x2)) = x1 + x2    40.21/19.01
POL(false) = 0    40.21/19.01
POL(le(x1, x2)) = 0    40.21/19.01
POL(minus(x1, x2)) = x1    40.21/19.01
POL(s(x1)) = [4] + x1    40.21/19.01
POL(true) = 0   
40.21/19.01
40.21/19.01

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.01
le(s(z0), 0) → false 40.21/19.01
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.01
minus(z0, 0) → z0 40.21/19.01
minus(0, z0) → 0 40.21/19.01
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
gcd(0, z0) → z0 40.21/19.02
gcd(s(z0), 0) → s(z0) 40.21/19.02
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.02
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.02
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c10, c8, c8, c9

40.21/19.02
40.21/19.02

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)))
We considered the (Usable) Rules:

minus(z0, 0) → z0 40.21/19.02
minus(0, z0) → 0 40.21/19.02
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
le(0, z0) → true 40.21/19.02
le(s(z0), 0) → false 40.21/19.02
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 40.21/19.02

POL(0) = 0    40.21/19.02
POL(GCD(x1, x2)) = [1] + [2]x1 + [2]x2    40.21/19.02
POL(IF_GCD(x1, x2, x3)) = x1 + [2]x2 + [2]x3    40.21/19.02
POL(LE(x1, x2)) = 0    40.21/19.02
POL(MINUS(x1, x2)) = 0    40.21/19.02
POL(c10(x1, x2)) = x1 + x2    40.21/19.02
POL(c2(x1)) = x1    40.21/19.02
POL(c5(x1)) = x1    40.21/19.02
POL(c8(x1)) = x1    40.21/19.02
POL(c8(x1, x2)) = x1 + x2    40.21/19.02
POL(c9(x1, x2)) = x1 + x2    40.21/19.02
POL(false) = [1]    40.21/19.02
POL(le(x1, x2)) = [1]    40.21/19.02
POL(minus(x1, x2)) = x1    40.21/19.02
POL(s(x1)) = [4] + x1    40.21/19.02
POL(true) = 0   
40.21/19.02
40.21/19.02

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.02
le(s(z0), 0) → false 40.21/19.02
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.02
minus(z0, 0) → z0 40.21/19.02
minus(0, z0) → 0 40.21/19.02
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
gcd(0, z0) → z0 40.21/19.02
gcd(s(z0), 0) → s(z0) 40.21/19.02
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.02
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.02
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_GCD, GCD

Compound Symbols:

c2, c5, c10, c8, c8, c9

40.21/19.02
40.21/19.02

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF_GCD(false, z0, z1) → c10(GCD(minus(z1, z0), z0), MINUS(z1, z0)) by

IF_GCD(false, 0, z0) → c10(GCD(z0, 0), MINUS(z0, 0)) 40.21/19.02
IF_GCD(false, z0, 0) → c10(GCD(0, z0), MINUS(0, z0)) 40.21/19.02
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
40.21/19.02
40.21/19.02

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.02
le(s(z0), 0) → false 40.21/19.02
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.02
minus(z0, 0) → z0 40.21/19.02
minus(0, z0) → 0 40.21/19.02
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
gcd(0, z0) → z0 40.21/19.02
gcd(s(z0), 0) → s(z0) 40.21/19.02
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.02
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.02
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
IF_GCD(false, 0, z0) → c10(GCD(z0, 0), MINUS(z0, 0)) 40.21/19.02
IF_GCD(false, z0, 0) → c10(GCD(0, z0), MINUS(0, z0)) 40.21/19.02
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(false, 0, z0) → c10(GCD(z0, 0), MINUS(z0, 0)) 40.21/19.02
IF_GCD(false, z0, 0) → c10(GCD(0, z0), MINUS(0, z0)) 40.21/19.02
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.02
40.21/19.02

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
40.21/19.02
40.21/19.02

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.02
le(s(z0), 0) → false 40.21/19.02
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.02
minus(z0, 0) → z0 40.21/19.02
minus(0, z0) → 0 40.21/19.02
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
gcd(0, z0) → z0 40.21/19.02
gcd(s(z0), 0) → s(z0) 40.21/19.02
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.02
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.02
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
IF_GCD(false, 0, z0) → c10 40.21/19.02
IF_GCD(false, z0, 0) → c10
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
IF_GCD(false, 0, z0) → c10 40.21/19.02
IF_GCD(false, z0, 0) → c10
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10, c10

40.21/19.02
40.21/19.02

(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

IF_GCD(false, z0, 0) → c10 40.21/19.02
IF_GCD(false, 0, z0) → c10
40.21/19.02
40.21/19.02

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.02
le(s(z0), 0) → false 40.21/19.02
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.02
minus(z0, 0) → z0 40.21/19.02
minus(0, z0) → 0 40.21/19.02
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.02
gcd(0, z0) → z0 40.21/19.02
gcd(s(z0), 0) → s(z0) 40.21/19.02
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.02
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.02
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.02
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.02
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.02
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.02
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.02
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.03
40.21/19.03

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:

minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 40.21/19.03

POL(0) = 0    40.21/19.03
POL(GCD(x1, x2)) = [4]x1 + [4]x2    40.21/19.03
POL(IF_GCD(x1, x2, x3)) = [4]x2 + [4]x3    40.21/19.03
POL(LE(x1, x2)) = 0    40.21/19.03
POL(MINUS(x1, x2)) = 0    40.21/19.03
POL(c10(x1, x2)) = x1 + x2    40.21/19.03
POL(c2(x1)) = x1    40.21/19.03
POL(c5(x1)) = x1    40.21/19.03
POL(c8(x1)) = x1    40.21/19.03
POL(c8(x1, x2)) = x1 + x2    40.21/19.03
POL(c9(x1, x2)) = x1 + x2    40.21/19.03
POL(false) = 0    40.21/19.03
POL(le(x1, x2)) = 0    40.21/19.03
POL(minus(x1, x2)) = x1    40.21/19.03
POL(s(x1)) = [2] + x1    40.21/19.03
POL(true) = 0   
40.21/19.03
40.21/19.03

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.03
minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
gcd(0, z0) → z0 40.21/19.03
gcd(s(z0), 0) → s(z0) 40.21/19.03
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.03
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.03
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.03
40.21/19.03

(25) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
40.21/19.03
40.21/19.03

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.03
minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
gcd(0, z0) → z0 40.21/19.03
gcd(s(z0), 0) → s(z0) 40.21/19.03
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.03
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.03
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0))))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.03
40.21/19.03

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 40.21/19.03

POL(0) = 0    40.21/19.03
POL(GCD(x1, x2)) = [3] + [2]x1·x2    40.21/19.03
POL(IF_GCD(x1, x2, x3)) = [2]x2·x3    40.21/19.03
POL(LE(x1, x2)) = 0    40.21/19.03
POL(MINUS(x1, x2)) = [3] + x2    40.21/19.03
POL(c10(x1, x2)) = x1 + x2    40.21/19.03
POL(c2(x1)) = x1    40.21/19.03
POL(c5(x1)) = x1    40.21/19.03
POL(c8(x1)) = x1    40.21/19.03
POL(c8(x1, x2)) = x1 + x2    40.21/19.03
POL(c9(x1, x2)) = x1 + x2    40.21/19.03
POL(false) = 0    40.21/19.03
POL(le(x1, x2)) = 0    40.21/19.03
POL(minus(x1, x2)) = x1    40.21/19.03
POL(s(x1)) = [2] + x1    40.21/19.03
POL(true) = 0   
40.21/19.03
40.21/19.03

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.03
minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
gcd(0, z0) → z0 40.21/19.03
gcd(s(z0), 0) → s(z0) 40.21/19.03
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.03
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.03
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.03
40.21/19.03

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

LE(s(z0), s(z1)) → c2(LE(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 40.21/19.03

POL(0) = 0    40.21/19.03
POL(GCD(x1, x2)) = [1] + [2]x2 + [2]x1·x2    40.21/19.03
POL(IF_GCD(x1, x2, x3)) = [1] + [2]x2·x3    40.21/19.03
POL(LE(x1, x2)) = [3] + [2]x1    40.21/19.03
POL(MINUS(x1, x2)) = [2]x2    40.21/19.03
POL(c10(x1, x2)) = x1 + x2    40.21/19.03
POL(c2(x1)) = x1    40.21/19.03
POL(c5(x1)) = x1    40.21/19.03
POL(c8(x1)) = x1    40.21/19.03
POL(c8(x1, x2)) = x1 + x2    40.21/19.03
POL(c9(x1, x2)) = x1 + x2    40.21/19.03
POL(false) = 0    40.21/19.03
POL(le(x1, x2)) = 0    40.21/19.03
POL(minus(x1, x2)) = x1    40.21/19.03
POL(s(x1)) = [2] + x1    40.21/19.03
POL(true) = 0   
40.21/19.03
40.21/19.03

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 40.21/19.03
le(s(z0), 0) → false 40.21/19.03
le(s(z0), s(z1)) → le(z0, z1) 40.21/19.03
minus(z0, 0) → z0 40.21/19.03
minus(0, z0) → 0 40.21/19.03
minus(s(z0), s(z1)) → minus(z0, z1) 40.21/19.03
gcd(0, z0) → z0 40.21/19.03
gcd(s(z0), 0) → s(z0) 40.21/19.03
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 40.21/19.03
if_gcd(true, z0, z1) → gcd(minus(z0, z1), z1) 40.21/19.03
if_gcd(false, z0, z1) → gcd(minus(z1, z0), z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:none
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(z0), s(0)) → c8(IF_GCD(true, s(z0), s(0))) 40.21/19.03
IF_GCD(false, s(z1), s(z0)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1))) 40.21/19.03
GCD(s(s(z1)), s(s(z0))) → c8(IF_GCD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1))) 40.21/19.03
GCD(s(0), s(s(z0))) → c8(IF_GCD(false, s(0), s(s(z0)))) 40.21/19.03
MINUS(s(z0), s(z1)) → c5(MINUS(z0, z1)) 40.21/19.03
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c8, c9, c10

40.21/19.03
40.21/19.03

(31) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
40.21/19.03
40.21/19.03

(32) BOUNDS(O(1), O(1))

40.21/19.03
40.21/19.03
40.21/19.09 EOF